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Chapter 32

Chapter 32. Inductance. Inductance. Self-inductance A time-varying current in a circuit produces an induced emf opposing the emf that initially set up the time-varying current. Basis of the electrical circuit element called an inductor Energy is stored in the magnetic field of an inductor.

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Chapter 32

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  1. Chapter 32 Inductance

  2. Inductance • Self-inductance • A time-varying current in a circuit produces an induced emf opposing the emf that initially set up the time-varying current. • Basis of the electrical circuit element called an inductor • Energy is stored in the magnetic field of an inductor. • There is an energy density associated with the magnetic field. • Mutual induction • An emf is induced in a coil as a result of a changing magnetic flux produced by a second coil. • Circuits may contain inductors as well as resistors and capacitors. Introduction

  3. Joseph Henry • 1797 – 1878 • American physicist • First director of the Smithsonian • First president of the Academy of Natural Science • Improved design of electromagnet • Constructed one of the first motors • Discovered self-inductance • Didn’t publish his results • Unit of inductance is named in his honor Section 32.1

  4. Some Terminology • Use emf and current when they are caused by batteries or other sources. • Use induced emf and induced current when they are caused by changing magnetic fields. • When dealing with problems in electromagnetism, it is important to distinguish between the two situations. Section 32.1

  5. Self-Inductance • When the switch is closed, the current does not immediately reach its maximum value. • Faraday’s law of electromagnetic induction can be used to describe the effect. • As the current increases with time, the magnetic flux through the circuit loop due to this current also increases with time. • This increasing flux creates an induced emf in the circuit. Section 32.1

  6. Self-Inductance, cont. • The direction of the induced emf is such that it would cause an induced current in the loop which would establish a magnetic field opposing the change in the original magnetic field. • The direction of the induced emf is opposite the direction of the emf of the battery. • This results in a gradual increase in the current to its final equilibrium value. • This effect is called self-inductance. • Because the changing flux through the circuit and the resultant induced emf arise from the circuit itself. • The emf εL is called a self-induced emf. Section 32.1

  7. Self-Inductance, Equations • An induced emf is always proportional to the time rate of change of the current. • The emf is proportional to the flux, which is proportional to the field and the field is proportional to the current. • L is a constant of proportionality called the inductance of the coil. • It depends on the geometry of the coil and other physical characteristics. Section 32.1

  8. Inductance of a Coil • A closely spaced coil of N turns carrying current I has an inductance of • The inductance is a measure of the opposition to a change in current. Section 32.1

  9. Inductance Units • The SI unit of inductance is the henry (H) • Named for Joseph Henry Section 32.1

  10. Inductance of a Solenoid • Assume a uniformly wound solenoid having N turns and length ℓ. • Assume ℓ is much greater than the radius of the solenoid. • The flux through each turn of area A is • The inductance is • This shows that L depends on the geometry of the object. Section 32.1

  11. RL Circuit, Introduction • A circuit element that has a large self-inductance is called an inductor. • The circuit symbol is • We assume the self-inductance of the rest of the circuit is negligible compared to the inductor. • However, even without a coil, a circuit will have some self-inductance. Section 32.2

  12. Effect of an Inductor in a Circuit • The inductance results in a back emf. • Therefore, the inductor in a circuit opposes changes in current in that circuit. • The inductor attempts to keep the current the same way it was before the change occurred. • The inductor can cause the circuit to be “sluggish” as it reacts to changes in the voltage. Section 32.2

  13. RL Circuit, Analysis • An RL circuit contains an inductor and a resistor. • Assume S2 is connected to a • When switch S1 is closed (at time t = 0), the current begins to increase. • At the same time, a back emf is induced in the inductor that opposes the original increasing current. Section 32.2

  14. RL Circuit, Analysis, cont. • Applying Kirchhoff’s loop rule to the previous circuit in the clockwise direction gives • Looking at the current, we find Section 32.2

  15. RL Circuit, Analysis, Final • The inductor affects the current exponentially. • The current does not instantly increase to its final equilibrium value. • If there is no inductor, the exponential term goes to zero and the current would instantaneously reach its maximum value as expected. Section 32.2

  16. RL Circuit, Time Constant • The expression for the current can also be expressed in terms of the time constant, t, of the circuit. • where t = L / R • Physically, t is the time required for the current to reach 63.2% of its maximum value. Section 32.2

  17. RL Circuit, Current-Time Graph, Charging • The equilibrium value of the current is e /R and is reached as t approaches infinity. • The current initially increases very rapidly. • The current then gradually approaches the equilibrium value. Section 32.2

  18. RL Circuit, Current-Time Graph, Discharging • The time rate of change of the current is a maximum at t = 0. • It falls off exponentially as t approaches infinity. • In general, Section 32.2

  19. RL Circuit Without A Battery • Now set S2 to position b • The circuit now contains just the right hand loop . • The battery has been eliminated. • The expression for the current becomes Section 32.2

  20. Energy in a Magnetic Field • In a circuit with an inductor, the battery must supply more energy than in a circuit without an inductor. • Part of the energy supplied by the battery appears as internal energy in the resistor. • The remaining energy is stored in the magnetic field of the inductor. Section 32.3

  21. Energy in a Magnetic Field, cont. • Looking at this energy (in terms of rate) • Ie is the rate at which energy is being supplied by the battery. • I2R is the rate at which the energy is being delivered to the resistor. • Therefore, LI (dI/dt) must be the rate at which the energy is being stored in the magnetic field. Section 32.3

  22. Energy in a Magnetic Field, final • Let U denote the energy stored in the inductor at any time. • The rate at which the energy is stored is • To find the total energy, integrate and Section 32.3

  23. Energy Density of a Magnetic Field • Given U = ½ L I2 and assume (for simplicity) a solenoid with L = mo n2 V • Since V is the volume of the solenoid, the magnetic energy density, uB is • This applies to any region in which a magnetic field exists (not just the solenoid). Section 32.3

  24. Energy Storage Summary • A resistor, inductor and capacitor all store energy through different mechanisms. • Charged capacitor • Stores energy as electric potential energy • Inductor • When it carries a current, stores energy as magnetic potential energy • Resistor • Energy delivered is transformed into internal energy Section 32.3

  25. Example: The Coaxial Cable • Calculate L of a length ℓfor the cable • The total flux is • Therefore, L is Section 32.3

  26. Mutual Inductance • The magnetic flux through the area enclosed by a circuit often varies with time because of time-varying currents in nearby circuits. • This process is known as mutual induction because it depends on the interaction of two circuits. Section 32.4

  27. Mutual Inductance, cont. • The current in coil 1 sets up a magnetic field. • Some of the magnetic field lines pass through coil 2. • Coil 1 has a current I1 and N1 turns. • Coil 2 has N2 turns. Section 32.4

  28. Mutual Inductance, final • The mutual inductance M12 of coil 2 with respect to coil 1 is • Mutual inductance depends on the geometry of both circuits and on their orientation with respect to each other. Section 32.4

  29. Induced emf in Mutual Inductance • If current I1 varies with time, the emf induced by coil 1 in coil 2 is • If the current is in coil 2, there is a mutual inductance M21. • If current 2 varies with time, the emf induced by coil 2 in coil 1 is Section 32.4

  30. Induced emf in Mutual Inductance, cont. • In mutual induction, the emf induced in one coil is always proportional to the rate at which the current in the other coil is changing. • The mutual inductance in one coil is equal to the mutual inductance in the other coil. • M12 = M21 = M • The induced emf’s can be expressed as Section 32.4

  31. LC Circuits • A capacitor is connected to an inductor in an LC circuit. • Assume the capacitor is initially charged and then the switch is closed. • Assume no resistance and no energy losses to radiation. Section 32.5

  32. Oscillations in an LC Circuit • Under the previous conditions, the current in the circuit and the charge on the capacitor oscillate between maximum positive and negative values. • With zero resistance, no energy is transformed into internal energy. • Ideally, the oscillations in the circuit persist indefinitely. • The idealizations are no resistance and no radiation. • The capacitor is fully charged. • The energy U in the circuit is stored in the electric field of the capacitor. • The energy is equal to Q2max / 2C. • The current in the circuit is zero. • No energy is stored in the inductor. • The switch is closed. Section 32.5

  33. Oscillations in an LC Circuit, cont. • The current is equal to the rate at which the charge changes on the capacitor. • As the capacitor discharges, the energy stored in the electric field decreases. • Since there is now a current, some energy is stored in the magnetic field of the inductor. • Energy is transferred from the electric field to the magnetic field. • Eventually, the capacitor becomes fully discharged. • It stores no energy. • All of the energy is stored in the magnetic field of the inductor. • The current reaches its maximum value. • The current now decreases in magnitude, recharging the capacitor with its plates having opposite their initial polarity. Section 32.5

  34. Oscillations in an LC Circuit, final • The capacitor becomes fully charged and the cycle repeats. • The energy continues to oscillate between the inductor and the capacitor. • The total energy stored in the LC circuit remains constant in time and equals. Section 32.5

  35. LC Circuit Analogy to Spring-Mass System, 1 • The potential energy ½kx2 stored in the spring is analogous to the electric potential energy (Qmax)2/(2C) stored in the capacitor. • All the energy is stored in the capacitor at t = 0. • This is analogous to the spring stretched to its amplitude. Section 32.5

  36. LC Circuit Analogy to Spring-Mass System, 2 • The kinetic energy (½ mv2) of the spring is analogous to the magnetic energy (½ L I2) stored in the inductor. • At t = ¼ T, all the energy is stored as magnetic energy in the inductor. • The maximum current occurs in the circuit. • This is analogous to the mass at equilibrium. Section 32.5

  37. LC Circuit Analogy to Spring-Mass System, 3 • At t = ½ T, the energy in the circuit is completely stored in the capacitor. • The polarity of the capacitor is reversed. • This is analogous to the spring stretched to -A. Section 32.5

  38. LC Circuit Analogy to Spring-Mass System, 4 • At t = ¾ T, the energy is again stored in the magnetic field of the inductor. • This is analogous to the mass again reaching the equilibrium position. Section 32.5

  39. LC Circuit Analogy to Spring-Mass System, 5 • At t = T, the cycle is completed • The conditions return to those identical to the initial conditions. • At other points in the cycle, energy is shared between the electric and magnetic fields. Section 32.5

  40. Time Functions of an LC Circuit • In an LC circuit, charge can be expressed as a function of time. • Q = Qmax cos (ωt + φ) • This is for an ideal LC circuit • The angular frequency, ω, of the circuit depends on the inductance and the capacitance. • It is the natural frequency of oscillation of the circuit. Section 32.5

  41. Time Functions of an LC Circuit, cont. • The current can be expressed as a function of time: • The total energy can be expressed as a function of time: Section 32.5

  42. Charge and Current in an LC Circuit • The charge on the capacitor oscillates between Qmax and -Qmax. • The current in the inductor oscillates between Imax and -Imax. • Q and I are 90o out of phase with each other • So when Q is a maximum, I is zero, etc. Section 32.5

  43. Energy in an LC Circuit – Graphs • The energy continually oscillates between the energy stored in the electric and magnetic fields. • When the total energy is stored in one field, the energy stored in the other field is zero. Section 32.5

  44. Notes About Real LC Circuits • In actual circuits, there is always some resistance. • Therefore, there is some energy transformed to internal energy. • Radiation is also inevitable in this type of circuit. • The total energy in the circuit continuously decreases as a result of these processes. Section 32.5

  45. The RLC Circuit • A circuit containing a resistor, an inductor and a capacitor is called an RLC Circuit. • Assume the resistor represents the total resistance of the circuit. Section 32.6

  46. RLC Circuit, Analysis • The total energy is not constant, since there is a transformation to internal energy in the resistor at the rate of dU/dt = -I2R. • Radiation losses are still ignored • The circuit’s operation can be expressed as Section 32.6

  47. RLC Circuit Compared to Damped Oscillators • The RLC circuit is analogous to a damped harmonic oscillator. • When R = 0 • The circuit reduces to an LC circuit and is equivalent to no damping in a mechanical oscillator. • When R is small: • The RLC circuit is analogous to light damping in a mechanical oscillator. • Q = Qmax e-Rt/2L cos ωdt • ωd is the angular frequency of oscillation for the circuit and Section 32.6

  48. RLC Circuit Compared to Damped Oscillators, cont. • When R is very large, the oscillations damp out very rapidly. • There is a critical value of R above which no oscillations occur. • If R = RC, the circuit is said to be critically damped. • When R > RC, the circuit is said to be overdamped. Section 32.6

  49. Damped RLC Circuit, Graph • The maximum value of Q decreases after each oscillation. • R < RC • This is analogous to the amplitude of a damped spring-mass system. Section 32.6

  50. Summary: Analogies Between Electrical and Mechanic Systems Section 32.6

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